There is an obvious product-free subset of the symmetric group of density 1/2, but what about the alternating group? An argument of Gowers shows that a product-free subset of the alternating group can have density at most n^(-1/3), but we only know examples of density n^(-1/2 + o(1)). We'll talk about why in fact n^(-1/2 + o(1)) is the right answer, why Gowers's argument can't prove that, and how this all fits in with a more general 'product mixing' phenomenon. Our tools include some nonabelian Fourier analysis and a Brascamp--Lieb-type inequality for the symmetric group due to Carlen, Lieb, and Loss.